Charged systems with density functional theory: Difference between revisions
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== Treating divergence in charge neutral calculations == | == Treating divergence in charge neutral calculations == | ||
VASP | VASP utilizes efficient fast Fourier transforms (FFT) to compute the electrostatic potential from the charge density using the Poisson equation, | ||
<math display="block"> | <math display="block"> | ||
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An obvious issue with computing <math display="inline">V(\mathbf{G})</math> is that it diverges for <math display="inline">\mathrm{G}\to 0</math>. | An obvious issue with computing <math display="inline">V(\mathbf{G})</math> is that it diverges for <math display="inline">\mathrm{G}\to 0</math>. | ||
This divergence is handled in charge neutral | This divergence is handled in charge neutral DFT calculations by cancelling out individual divergences for the total electrostatic energy,{{cite|ihm:jpcss:1979}} which is the sum of electron-electron, ion-electron and ion-ion energies | ||
<math display="block"> | <math display="block"> | ||
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</math> | </math> | ||
Individually, these terms have the following functional forms, | |||
<math display="block"> | <math display="block"> | ||
E_{\mathrm{electron-electron}} = \frac{1}{2} \sum_{\mathrm{G}} \rho_{\mathrm{ | E_{\mathrm{electron-electron}} = \frac{1}{2} \sum_{\mathrm{G}} \rho_{\mathrm{electron}}(\mathbf{G}) V_{\mathrm{electron}}(\mathbf{G}), | ||
</math> | </math> | ||
where <math display="inline">\rho_{\mathrm{ | where <math display="inline">\rho_{\mathrm{electron}}</math> and <math display="inline">V_{\mathrm{electron}}</math> are the electronic charge density and potential respectively. | ||
<math display="block"> | <math display="block"> | ||
E_{\mathrm{ion-electron}} = -\sum_{\mathrm{G}} \rho_{\mathrm{ion}}(\mathbf{G}) V_{\mathrm{ | E_{\mathrm{ion-electron}} = -\sum_{\mathrm{G}} \rho_{\mathrm{ion}}(\mathbf{G}) V_{\mathrm{electron}}(\mathbf{G}), | ||
</math> | </math> | ||
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<math display="block"> | <math display="block"> | ||
E_{\mathrm{electrostatic}}(\mathbf{G}=0) = \frac{1}{2}\rho_{\mathrm{ | E_{\mathrm{electrostatic}}(\mathbf{G}=0) = \frac{1}{2}\rho_{\mathrm{electron}}(\mathbf{G}=0) V_{\mathrm{electron}}(\mathbf{G}=0) - \rho_{\mathrm{ion}}(\mathbf{G}=0) V_{\mathrm{electron}}(\mathbf{G}=0) + \frac{1}{2}\rho_{\mathrm{ion}}(\mathbf{G}=0) V_{\mathrm{ion}}(\mathbf{G}=0), | ||
</math> | </math> | ||
which, can be expressed through the reciprocal space Poisson equation as, | which, can be expressed through the reciprocal space Poisson equation as, | ||
<math display="block"> | <math display="block"> | ||
E_{\mathrm{electrostatic}}(\mathbf{G}=0) = \frac{ | E_{\mathrm{electrostatic}}(\mathbf{G}=0) = \frac{4\pi}{\mathrm{G}^2} \left [\frac{1}{2}\rho_{\mathrm{electron}}(\mathbf{G}=0)^2 - \rho_{\mathrm{ion}}(\mathbf{G}=0)^2 + \frac{1}{2}\rho_{\mathrm{ion}}(\mathbf{G}=0)^2 \right]. | ||
</math> | </math> | ||
Under the condition that <math display="inline">\rho_{\mathrm{ | Under the condition that <math display="inline">\rho_{\mathrm{electron}}(\mathbf{G}=0)=\rho_{\mathrm{ion}}(\mathbf{G}=0)</math>, the individual divergences would cancel out and the total energy is convergent. | ||
Since the <math display="inline">\mathrm{G}=0</math> term of a Fourier transformed quantity is its average, | Since the <math display="inline">\mathrm{G}=0</math> term of a Fourier transformed quantity is its average, the above condition is satisfied when the average electron and ion charge density are the same, i.e. the system is charge neutral. | ||
== Divergences in charged calculations == | == Divergences in charged calculations == | ||
The total energy does not converge when <math display="inline">\rho_{\mathrm{ | The total energy does not converge when <math display="inline">\rho_{\mathrm{electron}}(\mathbf{G}=0) \neq \rho_{\mathrm{ion}}(\mathbf{G}=0)</math>, i.e., when the system is not charge neutral. | ||
Alternatively, we can think of this divergence as coming a compensating uniformly smeared out charge density, <math display="inline">\delta \rho</math> with energy <math display="inline">E_{\mathrm{compensating}}</math> | Alternatively, we can think of this divergence as coming a compensating uniformly smeared out charge density, <math display="inline">\delta \rho</math> with energy <math display="inline">E_{\mathrm{compensating}}</math> | ||
<math display="block"> | <math display="block"> | ||
E_{\mathrm{compensating}} = \frac{ | E_{\mathrm{compensating}} = \frac{4\pi}{\mathrm{G}^2} \left [ \delta \rho^2 \right] | ||
</math> | </math> | ||
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== Treating charged systems in practice == | == Treating charged systems in practice == | ||
We implement{{cite|vijay:arxiv:2024}} two separate methods to deal with charged systems. | |||
These methods are specific to the dimensionality of the system that is being computed. | These methods are specific to the dimensionality of the system that is being computed. | ||
Bulk charged systems in orthorhombic cells can be treated by | Bulk charged systems in orthorhombic cells can be treated by applying analytical monopole-monolpole corrections (see {{TAG|LMONO}} for further practical details). | ||
Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. | Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. | ||
These methods replace the Coulomb kernel <math display="inline"> v(\mathbf{G})</math>, i.e., the <math display="inline">\frac{4\pi}{\mathrm{G}^2}</math> in <math display="inline">V(\mathbf{G})</math> with a 2D kernel (<math display="inline"> v_{\text{2D}}(\mathbf{G})</math>) that removes electrostatic interactions between non-periodic replicas along the surface normal, | These methods replace the Coulomb kernel <math display="inline"> v(\mathbf{G})</math>, i.e., the <math display="inline">\frac{4\pi}{\mathrm{G}^2}</math> in <math display="inline">V(\mathbf{G})</math> with a 2D kernel{{cite|rozzi:prb:2006}} (<math display="inline"> v_{\text{2D}}(\mathbf{G})</math>) that removes electrostatic interactions between non-periodic replicas along the surface normal,{{cite|sohier:prb:2017}} | ||
<math display="block"> | <math display="block"> | ||
v_{\text{2D}}(\mathbf{G}) = | v_{\text{2D}}(\mathbf{G}) = | ||
\begin{cases} | \begin{cases} | ||
4\pi / \mathrm{G}^2 \left[ 1 - e^{-\mathrm{G}_{\parallel}R} \left ( \cos(\mathrm{G}_\perp R_\mathrm{c}) + \mathrm{G}_\perp / \mathrm{ | 4\pi / \mathrm{G}^2 \left[ 1 - e^{-\mathrm{G}_{\parallel}R} \left ( \cos(\mathrm{G}_\perp R_\mathrm{c}) + \mathrm{G}_\perp / \mathrm{G}_{\parallel} \sin(\mathrm{G}_{\perp}R_\mathrm{c}) \right ) \right] \quad & \mathrm{G} \neq 0 \\ | ||
4\pi / \mathrm{G}_\perp^2\left[ 1 - \cos(\mathrm{G}_\perp R_\mathrm{c}) - \mathrm{G}_{\perp}R_\mathrm{c} \sin(\mathrm{G}_\perp R_\mathrm{c}) \right] \quad & \mathrm{G}_{\parallel} = 0 \\ | 4\pi / \mathrm{G}_\perp^2\left[ 1 - \cos(\mathrm{G}_\perp R_\mathrm{c}) - \mathrm{G}_{\perp}R_\mathrm{c} \sin(\mathrm{G}_\perp R_\mathrm{c}) \right] \quad & \mathrm{G}_{\parallel} = 0 \\ | ||
-2\pi R_\mathrm{c}^2 \quad & \mathrm{G}=0 | -2\pi R_\mathrm{c}^2 \quad & \mathrm{G}=0 | ||
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</math> | </math> | ||
where <math display="inline">\mathrm{G}_{\parallel}</math> is the norm of the <math display="inline">\mathbf{ | where <math display="inline">\mathrm{G}_{\parallel}</math> is the norm of the <math display="inline">\mathbf{G}</math> vector parallel to the surface and <math display="inline">\mathrm{G}_{\perp}</math> is the <math display="inline">\mathbf{G}</math> vector along the surface normal. | ||
A similar change can be made to the Coulomb kernel to treat charged molecules (so-called 0D systems with <math display="inline"> v_{\text{0D}}(\mathbf{G}) to remove interactions with periodic replicas in all directions, | A similar change can be made to the Coulomb kernel to treat charged molecules (so-called 0D systems with <math display="inline"> v_{\text{0D}}(\mathbf{G}</math>)) to remove interactions with periodic replicas in all directions, | ||
<math display="block"> | <math display="block"> | ||
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\end{cases} | \end{cases} | ||
</math> | </math> | ||
See {{TAG|KERNEL_TRUNCATION/LTRUNCATE}} for further details on applying these Coulomb truncation methods in practice. | |||
== Related tags and articles == | |||
{{TAG|KERNEL_TRUNCATION/LTRUNCATE}}, | |||
{{TAG|KERNEL_TRUNCATION/LCOARSEN}}, | |||
{{TAG|KERNEL_TRUNCATION/IDIMENSIONALITY}}, | |||
{{TAG|KERNEL_TRUNCATION/ISURFACE}}, | |||
{{TAG|LMONO}} | |||
==References== |
Latest revision as of 10:35, 18 December 2024
On this page, we describe technical issues with computing the energies of charged systems with periodic density functional theory (DFT) calculations. We first introduce the problem of divergence of the electrostatic energy in DFT calculations and illustrate how this divergence is removed for charge neutral computations. We then discuss why this divergence is present for charged systems. Finally, we present methods which have been implemented in VASP that allow for calculations of charged bulk, surface and molecular systems.
Treating divergence in charge neutral calculations
VASP utilizes efficient fast Fourier transforms (FFT) to compute the electrostatic potential from the charge density using the Poisson equation,
where and are all points in real space. Fourier transforming the Poisson equation to reciprocal space,
where is the reciprocal lattice vector and is its norm.
An obvious issue with computing is that it diverges for . This divergence is handled in charge neutral DFT calculations by cancelling out individual divergences for the total electrostatic energy,[1] which is the sum of electron-electron, ion-electron and ion-ion energies
Individually, these terms have the following functional forms,
where and are the electronic charge density and potential respectively.
where is the ion charge density. VASP does not explicitly compute , but the potential of the ion is reflected through the eigenvalues. The ion-ion interactions are treated by Ewald summation
where is the only component which sums over the vectors and has the form,
The summed terms of , and are given by,
which, can be expressed through the reciprocal space Poisson equation as,
Under the condition that , the individual divergences would cancel out and the total energy is convergent. Since the term of a Fourier transformed quantity is its average, the above condition is satisfied when the average electron and ion charge density are the same, i.e. the system is charge neutral.
Divergences in charged calculations
The total energy does not converge when , i.e., when the system is not charge neutral. Alternatively, we can think of this divergence as coming a compensating uniformly smeared out charge density, with energy
which cancels out the divergent terms in .
The consequences of adding a uniform compensating charge is system specific. For highly screened systems, the potential coming from a moderate compensating charge presents an acceptable amount of error. Conversely, systems containing vacuum (such as 2D materials, surfaces and molecules) have little screening and hence are more susceptible to artifacts caused by this compensating charge.
Treating charged systems in practice
We implement[2] two separate methods to deal with charged systems. These methods are specific to the dimensionality of the system that is being computed.
Bulk charged systems in orthorhombic cells can be treated by applying analytical monopole-monolpole corrections (see LMONO for further practical details).
Surfaces and other two dimensional materials are treated with Coulomb kernel truncation methods. These methods replace the Coulomb kernel , i.e., the in with a 2D kernel[3] () that removes electrostatic interactions between non-periodic replicas along the surface normal,[4]
where is the norm of the vector parallel to the surface and is the vector along the surface normal. A similar change can be made to the Coulomb kernel to treat charged molecules (so-called 0D systems with )) to remove interactions with periodic replicas in all directions,
See KERNEL_TRUNCATION/LTRUNCATE for further details on applying these Coulomb truncation methods in practice.
Related tags and articles
KERNEL_TRUNCATION/LTRUNCATE, KERNEL_TRUNCATION/LCOARSEN, KERNEL_TRUNCATION/IDIMENSIONALITY, KERNEL_TRUNCATION/ISURFACE, LMONO
References
- ↑ J. Ihm, A. Zunger, M. L. Cohen, Journal of Physics C: Solid State Physics 12(21), 4409 (1979).
- ↑ S. Vijay, M. Schlipf, H. Miranda, F. Karsai, M. Marsman, G. Kresse, Manuscript in preparation (2024).
- ↑ C. A. Rozzi, D. Varsano, A. Marini, E. K. Gross, A. J. Rubio, Phys. Rev. B 73(20), 20511 (2006).
- ↑ T. Sohier, M. Calandra, and F. Mauri, Phys. Rev. B 96, 75448 (2017).